1. Global mean-first-passage time of random walks on Vicsek fractals
Wubin
Fudan University

2. Introduction

3. Fractals
Fractals are an important concept characterizing the features of real systems
We can obtain explicit closed-form solutions on fractals

4. Sierpinski gasket

5. T-fractal

6. Eigenvalues
Laplacian matrix
Eigenvalues

7. Discrete Random walks Assuming the time is discrete
The walker jumps uniformly from its current location to one of its neighbors

8. Global mean-first-passage time
First-passage time(FPT)
The expected time to hit a target node for the first time for a walker starting from a starting node
Global mean-first-passage time(GMFPT)
FPT averaged over all pairs of nodes

9. Vicsek fractals

10. Vicsek fractals

11. GMFPT General method Numerical result
Heavy demands on time and computational resources
Suit for small networks

12. GMFPT Using eigenvalues
Obtain the relation between GMFPT and network order directly

13. Comparison The relations between GMFPT and g
The filled symbols are the numerical results
the empty symbols correspond to the exact values

14. Comparison

15. Bound in Trees
The upper bound can be reached when the tree is a linear chain
The lower bound can be reached when the tree is a star graph

16. Conclusions
Using the connection between the FPTs and the Laplacian eigenvalues for general graphs, we have computed the GMFPT and obtained explicit solution
GMFPT grows approximately as a power-law function of N
The upper and lower bound for GMFPT can be achieved in linear chains and star graphs

18. Thank you